Development of a laser
interferometer for position
measurement
THESIS
submitted in partial fulfillment of the requirements for the degree of
BACHELOR OF SCIENCE in
PHYSICS
Author : Jan Nouws
Student ID : 1156071
Supervisor : Dr.Ir. S.J. Van der Molen
2ndcorrector : Prof.Dr. M. Van Exter
Development of a laser
interferometer for position
measurement
Jan Nouws
Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands
July 13, 2016
Abstract
In this research project, a step is made on the way to cryogenic low energy electron microscopy. This is done by determining the
feasibility of an optical method to measure the position of the 8 piezomotors controlling the motion stage to which the sample holder is attached. The idea is to use fiber interferometry to obtain
the desired position data. For this, a fiber interferometer is constructed. Measurements are performed and experiments are done, based upon which it is concluded that fiber interferometry
Contents
1 Introduction 3
1.1 Microscopy 3
1.2 The Low Energy Electron Microscope 4
1.3 CryoLEEM 7
1.3.1 Position measurement 9
2 Interferometry 10
2.1 Interference 10
2.2 The Michelson interferometer 11
2.3 The Fabry Perot interferometer 12
3 Fiber-interferometric position measurement 15
3.1 Fiber interferometers 15
3.2 Motion direction ambiguity 17
3.3 Modulation of the interferometer 17
3.3.1 Phase modulation 19
3.3.2 Frequency modulation 25
3.3.3 DFB lasers 26
3.3.4 Lock-in technique 27
3.4 Differentiate and cross multiply method 30
4 Experiments and Results 31
4.1 Introduction 31 4.2 Setup 31 4.2.1 Optics 31 4.2.2 Electronics 32 4.2.3 Software 33 4.3 Measurements 36
CONTENTS 2
4.3.1 Determining setup characterestics 36
4.3.2 Angle Dependence measurement 37
4.3.3 The modulated signal 38
4.3.4 Phasor diagram 38
4.3.5 Interferometry without a lens. 39
4.3.6 GRIN-lens measurements 41
4.3.7 Digital lock-in 45
4.4 Discussion 45
Chapter
1
Introduction
1.1
Microscopy
Since ancient times, mankind has been curious to explore and discover new, hidden phenomena in the universe. For example, to discover the phe-nomena which are too distant to be seen with the unaided eye, telescopes were designed, constructed and used. Another invisible world was dis-covered to exist at small length scales. To reveal the phenomena hidden there, mankind developed microscopes. The first microscopes, and still most of the present ones, use light which is focussed by a system of lenses to create a magnified image of the sample. However, due to the diffrac-tion limit, one cannot get proper images at length scales which are small compared to the wavelength of the light used, which is typically several hundreds of nanometers. To circumvent this restriction, electron micro-scopes were constructed. Electrons could be regarded as waves as well, but electrons usually have a much smaller wavelength, since the wave-length of electrons is given by:
λ= √ h 2meeV · 1 q 1+2meV ec2 (1.1)
Here λ is the wavelength of the electron, h is Planck’s constant, me is the mass of an electron, e is the elementary charge, V is the kinetic energy of the electrons, and c is the speed of light. Note that the right term is a relativistic correction and even at 15 kV this term is approximately equal to 1. This means that in a 15 kV electron microscope the wavelength is equal to 10pm, whereas in a 10 V device the wavelength is equal to 400pm.
1.2 The Low Energy Electron Microscope 4
1.2
The Low Energy Electron Microscope
Most electron microscopes use highly energetic electrons which are scat-tered by (SEM) or transmitted by (TEM) their targets. However, using lower energy electrons has some important advantages. For example, us-ing a low electron energy improves the surface sensitivity of the technique. A type of microscope that makes use of this principle is the Low Energy Electron Microscope (LEEM).[1]
The LEEM principle was already invented by Ernst Bauer in 1962. But the full development was not achieved until 1985, under the assistance of Wolfgang Telieps. Further development has been achieved by the efforts of Ruud Tromp. He also developed the basic geometry for the ESCHER (Electronic, Structural, and Chemical Nano-imaging in Real Time) setup in Leiden (see Figure 1.1), which includes the 90 degrees deflection of the electron path and the aberration correcting mirror seen in figure (1.2).[1]
Figure 1.1: The specific LEEM machine (ESCHER) involved in this research project.
As in other electron microscopes, the electrons are emitted from an electron gun, which, in the case of ESCHER, is on the top of the device. The electrons are accelerated downwards and collimated, after which their track is bent to one side due to the Lorentz force they experience from an applied magnetic field in magnetic prism array MPA1 (Figure 1.2). After
1.2 The Low Energy Electron Microscope 5
the curve transfer lens M1 focusses the beam into the back focal plane of a cathode objective lens. The high voltage between this lens and the sample decelerates the electrons, until they impinge the sample with an energy of 0-40 eV. After interaction with the sample, the same electric field causes the electrons to accelerate back towards the objective lens. A diffraction pattern is formed in the back focal plane of the objective lens and is trans-ferred to the entrance plane of MPA1. A real image exists on the diagonal of MPA1. After going through the projector column (P1 - P4 in Figure 1.2), the electrons form an image on a micro channel plate. The amplified im-age is projected on a phosphor screen which is filmed by a CCD camera, which is connected to a computer. P2 is the diffraction lens, it switches the entrance plane of the projector between an image plane and a diffrac-tion plane. This technique of imaging the diffracdiffrac-tion plane is called Low Energy Electron Diffraction (LEED). Another possibility in this set-up is to irradiate the sample with UV light and to detect the electrons that es-cape the sample by the photo electric effect, this is called Photon Emission Electron Microscopy (PEEM).[1] The samples in the LEEM are attached to a stage (see figures 1.3 and 1.4). This stage is equipped with 8 linear mo-tors to provide 5 degrees of freedom. To be as accurate as possible, piezo motors are used. These motors make use of piezo electric materials that change their shape under the influence of an electric field. One can find an example in figure 4.14. The requirements for the motors are at present:
• The motors should have a range of several cm.
• The motors should be able to apply a force on the stage of at least 5 N.
• The characteristics of the motors should not depend on the temper-ature.
• The motors should not be equipped with too many cables.
• The motors should be able to work under an ultra high vacuum and a high voltage.
Translation of the stage is possible in three dimensions, rotation in both horizontal directions (see Figure 1.4). This is achieved by attaching the stage on two motors, which enable the stage to move in one diagonal di-rection. This stage is attached to another stage with motors that move in the other diagonal direction. That stage is attached on four motors, with one motor on each corner of the stage. These four motors could be moved
1.2 The Low Energy Electron Microscope 6
Figure 1.2: A schematic overview of the path of the electrons in the LEEM. a) an uncorrected LEEM setup. The electrons come down from an electron gun until their path is bent by a Magnetic prism array, which can be thought of as a magnet. After reaching the sample the electrons are bent down towards the detector. b) The LEEM system in this research project. This device has an additional aberra-tion correcting mirror, for improving the resoluaberra-tion of the images.
[2]
simultaneously, enabling the stage to move in the vertical direction. One could also choose to move only two motors at one side, allowing the stage to tilt in the two possible directions. As can be seen in Figure 1.4), the motors at the top of the stage enable the stage to move in both horizontal directions, whereas the motors at the feet of the stage enable the stage to tilt, or move in the vertical direction. [1]
1.3 CryoLEEM 7
1.3
CryoLEEM
The samples in the LEEM can already be observed at a wide range of tem-peratures extending from 300 to 1800 K. However, since there are expected to exist interesting phenomena at temperatures outside of this range, it has been investigated whether the temperature range could be extended. Es-pecially at lower temperatures it is expected that phenomena are waiting to be studied. Therefore, the ultimate goal of this research is to decrease the lower boundary of the temperature range to about 10 K, and to build a separate stage for measurements at temperatures below 300 K.[3][1]
1.3 CryoLEEM 8
Figure 1.4: The intended stage for the LEEM to be used in the new, ”cold” side. The motors a, b, c and d are used for moving the stage in the horizontal plane, The motors 1, 2, 3 and 4 are used for tilting the stage, or move the stage in the vertical direction.
Figure 1.5: The piezomotors as they are used in the LEEM. The middle part, the rotor, can be moved independently of the rest of the motor, the stator.
1.3 CryoLEEM 9
1.3.1
Position measurement
For accurate control of the stage, it is necessary to accurately know the po-sition of the motors within their range. One way to do this, is to use an optical ruler, made of reflecting patterns. By looking at those reflections one can find the position. However, on the ”cold” side, where the sample can be cooled down to 10K and the motors must be cooled as well in or-der to reduce the heat transfer to the sample and to minimize the effects of outgassing, complex electronics will be unreliable. Also, this method requires a lot of cables, which would increase the heat transfer to the mo-tors. Another possible method is to form a capacitor between the fixed and the moveable parts of the motor. In this case, the amount of overlap determines the capacitance, which is a measurable quantity. This method requires less wiring but is also less accurate. A third possible method is to attach a mirror on the movable part, and use interferometry to measure the position. This seems to be a reasonable option, because the only things that have to be put in the cryogenic environment are a piece of fiber, a lens and a mirror, the other parts can be set up outside the cryogenic environ-ment. This specific research project aims to determine the feasibility of this method.
Chapter
2
Interferometry
2.1
Interference
Interference is known as the phenomenon that when two waves are com-bined, a resulting wave occurs, with an irradiance that depends on the phase difference of the separate waves. In general the fields add up and the irradiance, which is the square of the field, is given by:
I = I1+I2+2pI1I2cosφ (2.1)
Here I1and I2are the intensities of the two light beams and φ is the phase difference between them.[4] This means that if the waves are exactly in phase, the resulting wave will be stronger. If, on the other hand, the phase difference is exactly π radians, the resulting wave will have an irradiance which is decreased, or even, if the incoming waves have the same irradi-ance, fully extinct. Measuring interference is usually easier if the differ-ence between the maximal and the minimal value is large. To express this more quantitatively, the fringe contrast C is introduced, which is given by:
C = Imax−Imin
Imax (2.2)
where I is the measured intensity.[5] A fringe contrast of 1 corresponds to a signal where the minimal values are zero, ensuring a maximal contrast be-tween the maximal and minimal values. A fringe contrast of 0 corresponds to a signal without maxima and minima. Although interference is most well-known when it is related to light, the principle also works for other types of waves. The device which measures wave interference is called an interferometer. Normally, an interferometer splits a light beam into
2.2 The Michelson interferometer 11
Figure 2.1: The typical setup of a Michelson interferometer. Interference occurs due to the fact that the two paths may have a different length. The path lengths can be modified by moving the mirrors.[6]
several beams, and after going through different optical paths, the beams are recombined. Where the path difference originates from depends on the type of the interferometer. Interferometers come in quite a number of different types, but two of them are particularly interesting for position measurements. They will be described hereafter.
2.2
The Michelson interferometer
One of the most well known interferometer types is the Michelson inter-ferometer. The Michelson interferometer uses a light source, typically a laser because of the large coherence length, shining onto a beamsplitter. The beamsplitter is a mirror which reflects about fifty percent of the in-coming light, leading to a reflected and a transmitted part (see Figure 2.1). Since the beamsplitter is put at an angle of 45 degrees, the reflected part goes off to one side. Now both beams are reflected by a mirror, back into the beamsplitter. Now, a part of the reflected beam will transmit and a part of the transmitted beam reflects, and these two parts form an interfer-ence pattern, which depends on the differinterfer-ence in the path lengths of the transmitted and the reflected beam.
2.3 The Fabry Perot interferometer 12
2.3
The Fabry Perot interferometer
The conventional Fabry-Perot interferometer comprises a light source, which illuminates two parallel reflecting surfaces. The space between the sur-faces is called the etalon. To detect the light, one could use a detector which may be put on both sides of the etalon, depending on whether one would like to measure the reflectivity or the transmittance. Interference occurs due to the fact that light can undergo multiple reflections in the etalon, leading to different path lengths of the light that has experienced a different amount of reflections (see figure 2.2), which leads to a phase difference. Typically one inserts two collimating lenses outside of the sur-faces to converge the different beams. If assumed that x is the width of the etalon, n is the refractive index and θ is the angle of inclination, the path difference between two adjacent rays l is given by:
l=2nx cos(θ) (2.3)
The reflectivity of the surface makes quite a difference. The usual quantity to measure this is the finesse coefficient. Note that, if one thinks about re-flectivity, one can distinguish the relative amplitude of the reflected wave (r) or the relative reflected power (R), where R is equal to|r|2. The finesse coefficient can be calculated using the following formula:
F=4 √ R1R2 1−R1R2 (2.4) Here R1 and R2 are the reflectivities of the surfaces. In most etalons the two surfaces are made of the same material and have therefore the same reflectivity. Note that the reflectance varies between 0 and 1, correspond-ing to a value of 0 or infinity for the finesse coefficient. One can also define the finesseΦ, related to the finesse coefficient with the following equation: [4]
Φ= π
2arcsin √1 F
(2.5)
The transmittance of an Fabry-Perot interferometer is given by an Airy Function, which is given by:
T = 1
1+Fsin2(φ) (2.6)
One could also put the detector at the same side as the light source, in that case the reflectivity is measured instead of the transmittance, The reflec-tivity is equal to 1 minus the transmittance, but this can also be translated
2.3 The Fabry Perot interferometer 13
Figure 2.2:An example of an etalon, used in a regular Fabry-Perot interferometer. One typically uses a lens to focus the transmitted beams on a detector. One could change the reflectivity of the surfaces, varying the finesse coefficient.[15]
into the following equation:
R = Fsin
2( φ)
1+Fsin2(φ) (2.7)
where φ is the phase difference between the light beams. This function is shown in figure 2.3. One could derive these functions by converting the incident wave to its complex representation E0eiωt. Assuming that E0 is the amplitude of the incoming wave, EN the amplitude of the wave after experiencing N reflections, t is the transmittance and r is the reflectance, and t0 and r0represent the transmittance and the reflectance coming from inside the etalon (see fig.2.2),
E1= E0reiωt E2=E0tr0t0ei(ωt−δ)
E3 =E0tr03t0ei(ωt−2δ)
EN =E0tr02N−3t0ei(ωt−(N−1)δ)
(2.8)
The δ occurs due to the phase change of the waves, and is given by 4πlλ . The final intensity is given by the infinite sum of the terms above, which converges to, if one assumes that r = −r0and tt0 =1−r2(which is correct if the absence of any absorption is provided):
ER =E0eiωtr(1−e −iδ)
(1−r2)eiδ
(2.9) The intensity is equal to the scalar amplitude of the waves, multiplied by its complex conjugate, Therefore R is given by:
R= E
2
0r2(1−eiδ)
2.3 The Fabry Perot interferometer 14
Since eiδ+e−iδ =2cosδ and, if a new variable φ= δ
2 is introduced, cos δ is equal to 1−sin2φ, this can be transformed to:
R= I0 ( 2r
1−r2)sin2φ
1+ (1−2rr2)sin2φ
(2.11) And since the finesse coefficient for two equally reflecting surfaces is given by(1−2rr2)2, this is exactly equal to the reflective airy function. If the finesse
is small enough, the Airy function is approximately equal to a sinusoidal wave. If, however the finesse is very high, the Airy function is approxi-mately equal to zero, except at phase differences which are similar to 2π, where it will equal the initial irradiance.[4]
Figure 2.3:A plot of four reflectivity functions, with different finesse coefficients. The different finesse coefficients are 0.1, 1, 10, and 100 from bottom to top. The lower the finesse coefficient, the more the function will look like a sinusoidal wave, since the denominator is approximately equal to 1 for small values of F.
Chapter
3
Fiber-interferometric position
measurement
3.1
Fiber interferometers
Since it is necessary to build quite a number of interferometers (eight of them are needed for the LEEM) and since the ”cold” stage demands a cryo-genic environment, where regular optical setups easily misalign, it makes more sense to use fiber interferometers, and that would bring on a few changes compared to regular interferometers. The most important one is the beamsplitter, which is replaced by a fibersplitter. A fibersplitter con-sists of a single mode fused coupler in which the evanescent fields strongly interact, forming an x-shaped piece of fiber. The splitter, often called a -3db-coupler (3db corresponds to a factor 2), should divide the light irra-diance equally over the two fiber arms in the same direction. This type of Michelson is shown in Figure 3.1. At one side of the splitter there are the light source and the detector. At the other side there are two mirrors. The idea is to make sure that one of the mirror arms is fixed in length. This arm is used as a reference arm to generate the interference pattern with the other arm, and the length of the other arm contains the length that is to be measured measured. Unfortunately, this brings up one disadvantage of this interferometer. The two light beams go along different paths, and therefore there could be changes in these paths, for example due to heat-ing and coolheat-ing of the fiber. This could lead to length differences which are measured, but are not intended to be measured, resulting in systematic er-rors. One way to avoid the problem of length differences, is to make use of the Fabry-Perot interferometer, that is, to make sure that there are no path
3.1 Fiber interferometers 16
Figure 3.1: The global setup of the fiber version of a Michelson interferometer. The laser (orange) emits light, The light goes through the fiber splitter (blue) and is reflected at the mirrors (M1 and M2) After reflecting the light goes back to the fibersplitter and further to the detector (green).
Figure 3.2:The global setup for the Fabry-Perot interferometer. The laser (below) emits light, The light goes through the fibersplitter (FC). At the end of the fiber (at the right), the laser light (red) escapes the fiber and is reflected by a mirror back into the fiber. The the light goes through the beam splitter again before it arrives at the detector (left).
differences at the fiber parts of the interferometer. The idea here is to put a single mirror behind the end of the fiber. Now part of the light already reflects from the fiber surface, while the rest of the light goes through the surface and might be reflected at the mirror, before being coupled back into the fiber see figure 3.2. The amount of light which reflects at the fiber surface can be calculated using:
R=nt−ni
nt+ni 2
[4] (3.1)
Here ni and nt are the refractive indexes of the fiber and the material out-side the fiber (normally air) and R is the reflectivity. Since the refractive index of air is about 1 and from the specifications it is known that the re-fractive index of the fiber is about 1.65356, the reflectivity will be about
3.2 Motion direction ambiguity 17
6,066%. At high finesse, it is hard to determine the position, since the dif-ferences in irradiance are most of the time too small. At low finesse, the response function is more like a sine wave, therefore the low finesse is actually more favorable (see figure 2.3). A problem of this type of interfer-ometer might be, although the same problem arises using the Michelson type, the quite small angle at which the reflected light is no longer cou-pled back into the fiber. This angle can be calculated using the following equation.
α =
arctan(df)
2 (3.2)
Here d is the radius of the fiber core and f the focal length of the collimating lens. Knowing that d is about 9µm and f is about 4mm, the critical angle is about 0.01 degrees. [5]
3.2
Motion direction ambiguity
If one succeeds in measuring the intensity, one can still not uniquely de-termine the phase change after displacement of the mirror. If, for example, the irradiance goes up, while it was initially at the minimal value, the mir-ror could have moved in both possible directions. Also, the sensitivity of the interferometer at the signal extrema is zero. An important aim of this project is to turn a fiber interferometer, which is seen to be a poor long range position sensor, into a very good one. Given the number of sensors needed it is important that the complexity remains reasonable and the cost remains low.
3.3
Modulation of the interferometer
The direction ambiguity of the laser interferometer can be avoided and the sensitivity can be made more uniform by measuring the intensity while the phase φ of the interferometer is changed in a periodic manner. This can be done by the closely related concepts of phase modulation (PM) and frequency modulation (FM). We will define phase modulation as the case where the mirror is moved to change φ and frequency modulation as the case where the laser frequency ν = c/λ is changed. The data processing for these two methods will turn out to be very similar. The idea is that by modulating the interferometer phase φ it becomes possible to obtain not just an intensity I but also an intensity Q where I and Q are in phase quadrature. I and Q are called a quadrature pair and the procedure of
3.3 Modulation of the interferometer 18
obtaining such a pair is called quadrature demodulation. The orthogonal (90 degrees out of phase) components I and Q can be plotted in the com-plex plane to form a phasor. This phasor will rotate over a circle when the mirror is moved, thereby solving the direction ambiguity problem: when the phase is increased, the phasor rotates one way and when the phase is decreased it rotates the opposite way. The phase is simply found from:
φ=atan2(I, Q). (3.3)
The atan2 function can be used to compute the phase (the argument) in the complex plane. It cancels the problem that the usual arctangent func-tion yields the same value for arguments which are 180◦ out of phase. Its behavior is described in table 3.1.
atan2(x, y) x <0 x =0 x>0 y>0 arctanyx+π π2 arctan y x y=0 π ∅ 0 y<0 arctanyx−π −2π arctan y x
Table 3.1: The behavior of the atan2 function. x and y are the real and the imag-inary part respectively. Using this function, the calculated phase varies from−π
to π, where, at the negative real axis, the argument value is set to π by convention. [7]
It should be understood that whenever the phase rotates through zero, depending on the direction 2π should be added to or subtracted from the phase. This ‘phase clock’ is directly related to the mirror position via:
x= φ
k (3.4)
where k is the ‘spatial frequency’ of the particular interferometer. [7] Given this straightforward procedure to obtain the position from a quadrature pair, the question is how to obtain the quadrature pair itself. Since we deal with frequency and phase modulation it makes sense to investigate the side bands of the signal.
3.3 Modulation of the interferometer 19
3.3.1
Phase modulation
The output intensity of an interferometer is periodic over a distance a. For a singly folded interferometer the light goes forth and back once and hence
a= λ
2 (3.5)
We have already seen that the intensity at the detector will be
I(x) = I1+I2+2pI1I2cos(kx) (3.6) where
k = 2π
a (3.7)
and x is the mirror position. Allowing for offsets due to an imperfect fringe visibility and normalizing the intensity to 1 we find that
I(x) = 1+cos(kx) (3.8)
I can be seen as the intensity or as the photodiode current, which for an InGaAs detector in the infrared is roughly equal to the intensity since the sensitivity of this detector is of the order of 1 mA/mW. Given the cosine output of the interferometer, the quadrature signal we are looking for is of the form
Q(x) = sin(kx) (3.9)
The idea of phase modulation is to move the mirror with a small am-plitude X around the equilibrium position x. For a sinusoidal phase mod-ulation with angular frequency ω the mirror position x then becomes x+
X sin(ωt). Substituting this in (3.8) results in the output signal of a phase modulated interferometer:
I(x, t) = 1+cos k(x+X sin(ωt)) (3.10) The modulation depth β is seen to be
β=kX (3.11)
(3.10) shows that the phase modulated interferometer is mathemati-cally similar to FM radio modulation without a carrier wave. For a small modulation depth β the signal will be confined to the first few side bands. To calculate the amplitude of these harmonics the Fourier transform of (3.10)
3.3 Modulation of the interferometer 20
must be calculated. We start by expanding the cosine expression using the identity cos(α+β) =cos α cos β−sin α sin β:
I(x, t) =1+cos(kx)cos(βsin(ωt)) −sin(kx)sin(βsin(ωt)) (3.12) By converting to complex exponentials and evaluating the Taylor ex-pansion of eiβsin(θ) the Fourier integral of this function can be calculated
which after a lengthy derivation yields the two identities: cos(βsin(θ)) = J0(β) +2 ∞
∑
m=1 J2m(β)cos(2mθ) sin(βsin(θ)) =2 ∞∑
l=0 J2l+1(β)sin((2l+1)θ) (3.13)where the functions Ji are the Bessel functions of the first kind. Combin-ing (3.12) and (3.13) gives the frequency response of the phase modulated interferometer: I(x, t) = 1+cos(kx)J0(β) +2 ∞
∑
k=1 J2k(β)cos(2kωt) −sin(kx)2 ∞∑
l=0 Jl+1(β)sin((2l+1)ωt) (3.14)It can be seen that a quadrature pair can indeed be obtained from these harmonics: the cosine term is multiplied only by even harmonics and the sine term is multiplied only by odd harmonics. The most basic quadrature phase demodulator based on this principle is clearly the J0, J1, J2 demod-ulator. When higher order Bessel functions are ignored, the output of the interferometer is obtained from (3.14) for k =1 and l=0:
I(x, t) =1+J0(β)cos(kx)
−2J1(β)sin(kx)sin(ωt)
+2J2(β)cos(kx)cos(2ωt)
(3.15)
[8] There exist several methods to demodulate such a signal. Since a local oscillator is available and the noise level is expected to be important, a lock-in technique is a good choice. This method will make use of two reference signals: reference signal 1 at frequency ω1 = ω with amplitude r1 and reference signal 2 at frequency ω2 = 2ω and amplitude r2 that must be phase shifted by 90 degrees. When these reference signals are
3.3 Modulation of the interferometer 21
Figure 3.3:Bessel functions of the first kind J0, J1and J2.
each mixed with I(x, t) and amplified with gains g1 and g2 respectively, the signals s01and s02are obtained:
s01 = −2g1r1J1(β)sin(kx)sin2(ωt) s02 =2g2r2J2(β)cos(kx)cos2(2ωt)
(3.16) Since cos2(x) = (cos(2x) +1)/2 and sin2(x) = (−cos(2x) +1)/2 these signals can be low pass filtered below the double frequency of their re-spective reference signal. Our J0, J1, J2demodulator thus produces the fol-lowing signals:
s0=1+J0(β)cos(kx) s1= −g1r1J1(β)sin(kx) s2=g2r2J2(β)cos(kx)
(3.17)
It can be concluded that this scheme indeed implements the idea of (3.3): tan(kx) = sin(kx) cos(kx) = − g2r2J2(β) g1r1J1(β) s1 s2 (3.18) It should be noted that although s0 did not play a role in the determina-tion of the posidetermina-tion, it is a DC offset to the first gain stage of the photodiode transimpedance amplifier. Care should be taken not to let this signal sat-urate that stage. A second important point to note is that the first term
3.3 Modulation of the interferometer 22
on the right hand side g2r2J2(β)
g1r1J1(β) of (3.18) should be made unity. If not,
the I, Q phasor will sweep out an ellipse rather than a circle. This is eas-ily corrected in software but the effect should not result in such unequal gains that ADC resolution is lost. We will see that the ratio J2/J1will play a more important role in frequency modulation. One should also consider that this quadrature method is designed to use with sine wave signals. Since the transfer function of the interferometer (see Figure 2.3) is not ex-actly sinusoidal, it is interesting to check the deviation error compared to a wave of the signals. One could compare the function with a sine wave with an identical period, amplitude and average. For a finesse coefficient F the signal is given by:
Fsin2(φ)
1+Fsin2(φ) (3.19)
Let’s call this signal J. This signal is compared with the following sine wave:
F 2F+2−
F
2F+2cos(2x) (3.20)
Let’s call this signal M. To measure the error it is preferred to measure the ”relative absolute error” J−FM
F+1
The absolute error, S, can be derived by rewriting the equation:
S = F 2 − F2 ·cos(2x) 1+ F2 −F2 ·cos(2x) − F 2F+2 − F 2F+2·cos(2x) (3.21) Multiplying J and M by the denominator of J:
S· (1+ F 2 − F 2 ·cos(2x)) = F 2 − F 2cos(2x) − F 2F+2 + 1 2F2 2F+2 + 1 2F2 2F+2·cos(2x) + F 2F+2·cos(2x) + 1 2F2 2F+2− 1 2F2 2F+2 ·cos 2( 2x) (3.22) Changing the denominator in J:
S· (1+ F 2 − F 2 ·cos(2x)) = F2+F 2F+2 − F2+F 2F+2 ·cos(2x) − F 2F+2+ 1 2F2 2F+2 + 1 2F2 2F+2 ·cos(2x) + F 2F+2·cos(2x) + 1 2F2 2F+2·cos(2x) − 1 4F2 2F+2− 1 4F2 2F+2 ·cos(4x) (3.23)
3.3 Modulation of the interferometer 23
A lot of terms will cancel out, and the final error is given by:
S = F
2 8F+8 −
F2
8F+8·cos(4x) (3.24)
This means that the maximum error is given by 4FF+24. If one normalizes this error by dividing the error by the amplitude of the functions (1+FF), one obtains: S= F 2(F+1) F(4F+4) = F3+F2 4F2+4F = F 4 (3.25)
And since F = 1−4RR2, the normalized error is given by 1−RR2, which is
ap-proximately equal to R for small values of R. The relative absolute error is shown in figure 3.4.
Figure 3.4:The absolute error of the transfer function. This function is created by subtracting the signal M from the signal J. The maxima of the error graph have the value of 4FF+24.
Another way to check the deviation is to apply the atan2 function on the signal J itself and a corresponding signal at π
4 out of phase(see figure 3.5. If these two signals form a perfectly uniform phasor rotation, the atan2 function should form a straight line, the deviation compared to that line may be used to check the deviation error. This is shown in figure 3.6. As in the first error derivation, it turns out that the maximal deviation error is quite large (about the same value as the reflectivity, and therefore one might need to apply an external correction on the signal.
3.3 Modulation of the interferometer 24
Figure 3.5: The atan2 function applied on the transfer function. One can find the discontinuous steps of the function due to the change in argument if the argument goes through the negative real axis. Note that the other parts of the function do not exactly correspond to a straight line.
Figure 3.6:The deviation of the transfer function. The shown picture corresponds to the deviation of a straight line of the graph in figure 3.5. The error is found to be quite large, and therefore, applying an external software correction is required.
3.3 Modulation of the interferometer 25
3.3.2
Frequency modulation
It has been shown that PM results in a vast improvement of the readout of an interferometer. A big disadvantage of the method is the requirement to move a mirror, especially in a cryogenic environment. In a typical im-plementation PM will at least require a piezoelectric element and a pair of heat conducting, electrically radiating wires. For the cryogenic environ-ment it would be an enormous improveenviron-ment if only a thin single mode fiber would have to be fed into the UHV. With the advent of low cost dis-tributed feedback telecom lasers with a reasonably large modulation without mode hopping this idea can hopefully be made a reality.
Let us extend the theory from the previous section to FM. PM started off with the idea that a phase difference ∆φ is generated by a mirror dis-placement∆x. Since φ=kx and thus∆φ=k∆x+x∆k, PM modulates the
phase by changing the first term, and it is clear that the second term is the one that is important for FM. To be clear: the way we have defined things, frequency modulation means modulating the phase by changing the laser frequency, not that we convert an optical frequency into the baseband of an electronic circuit! For the singly folded interferometer it has been shown in the previous section that k = 4πλ . For a small laser modulation∆λ the phase change, and thus the modulation depth is
β= dφ dλ∆λ = d(kx) dλ ∆λ= x dk dλ∆λ= −4πx λ2 ∆λ (3.26) It is clear that with the typical plasma mode modulation of 5 pm/mA for a low cost IR laser the high frequency modulation would be maximum 50 pm on a wavelength of 1.5 micron. In that case the laser diode mod-ulation current is 10-20% of the drive current. For a system dimension of 10 mm, equation (3.26) still gives 2.8 radians modulation depth, which is even above the J1/J2 crossover. This means that this method is entirely feasible in small systems.
It is very important to note that with FM, β is a function of x. This means that for large motion ranges an active form of correction for the J2/J1 Bessel function ratio is required. Figure 3.3 shows that for β ≈ 2.6, J1 = J2 but for β = 3.7 it is seen that J1 = 0 so that the in-phase sensitivity vanishes! The best approach to solve this problem is to provide a servo that controls β such that the amplitude of s1and s2are kept equal. The output of this servo is a measure of the absolute position. An alternative approach to determine the absolute position could be to use s0. [8]
3.3 Modulation of the interferometer 26
Figure 3.7:An example of the graphs one typically obtains after modulation. The precise shape of the graph could vary and tells something about the phase, but all the possible graphs are of the form cos(sin(x)).
3.3.3
DFB lasers
The laser type which is used is a distributed feedback laser (DFB laser) This type is chosen because it enables one to apply frequency modulation. A laser amplifies its emitted light using an optical cavity, where light is reflected between two mirrors, enabling the laser to emit at several cavity modes. In a DFB laser, the grating ensures the amplification of a single mode. In this type of laser, the frequency of the light which is emitted by the laser will change if the temperature in the laser is changed. This is in part due to thermal expansion of the grating and the entire cavity, which changes the reflected wavelength. In this experiment it is intended to ap-ply a periodic modulation on the laser current, and thus the changes in frequency are also expected to be periodic. Note that the grating will heat faster compared to the entire cavity. So the temperature response of this effect will also be less slow. There is a third occurring effect: A DFB laser is a semiconductor, and a change in current also changes the refractive in-dex of the gain medium. This effect occurs almost immediately. A DFB laser indeed is usually observed to have 2 thermal time constants, one in the ms range and one in the 0.1 ms range. For faster modulation the so called current modulation depth, typically 3-5 pm/mA, is the relevant num-ber. The main disadvantage of frequency modulation using a DFB laser
3.3 Modulation of the interferometer 27
is that the irradiance of the light source also changes, which slightly per-turbs the measurement, although this can be corrected for. In general it is desired to keep the modulation current of the laser as low as possible. Although commercially available lasers with a built-in modulation input exist, regarding the intended number of sensors to use, a laser with a sin-gle current connection, without a modulation input, is used in order to keep the cost low.
3.3.4
Lock-in technique
Figure 3.8: In this scheme, one could see how the signal is modulated and de-modulated again. The voltage signal coming from the IV converter is multiplied separately by two sine wave signals with a certain frequency f and 2f respectively. Then the signals are low-pass filtered, yielding the component of the signal with the same frequency as the two sine wave multipliers. Then the component values are plotted in the xy plane. The idea is that the plotted values should form an ellipse.
Once the quadrature signals are obtained after modulation of the sig-nal (see figure3.7, the sigsig-nal need to be demodulated again to get a result suitable for position measurement. A commonly used way to do this is to use a lock-in amplifier, to apply a low-pass filter, and to plot the obtained signals. The way this is done is shown in figure 3.8.
3.3 Modulation of the interferometer 28
The lock-in amplifier
The lock-in amplifier is a device that measures the phase difference of the component of a specified frequency of the signal with a sine wave of the same frequency. To do so, one can multiply the signal with the sine wave or its sign. The mechanism makes use of the fact that if one multiplies two sine waves, the product will automatically average to zero unless the sine waves have the same frequency. Whereas if this is the case, the product may contain a direct current component, which is at its maximum if the waves are in phase. If the phase difference is π radians, the direct current component will be at its minimum, and if the phase difference is π
2, the component will be equal to zero. To measure the direct current compo-nent, one could apply a low pass filter on the signal. Normally, a lock-in amplifier has a built-in low-pass filter. However, since the lock-in ampli-fier is quite a large device, it might be the case that this device is not very suitable for the final position measurement. One could try to fabricate a digital substitute for the lock-in device. Digitally multiplying the signal with a sine wave is quite simple, and to apply a low-pass filter, one could try to construct a digital filter.
The Finite Impulse Response (FIR) filter
A specific kind of a digital filter is known as the FIR filter, where FIR stands for ”finite impulse response”. The filter algorithm makes use of an one di-mensional array of a certain length, with certain coefficients. The idea is to replace each value in the data by the sum of a certain amount of previ-ous data. Each data point is multiplied by the corresponding coefficient. The coefficients then determine the specifications of the filter, for example, whether it is a low-pass or a high-pass filter, its cut-off frequency, and the order of the filter. The typical transfer function of an FIR low-pass filter is shown in figure 3.9.
One could calculate the coefficients which one should use analytically. One way to do so is to make use of the Kaiser-Bessel algorithm. This algo-rithm is designed for filters which have an odd order, which corresponds directly to an odd coefficient array length, and have an minimal attenua-tion of 21dB at some point. Using this method, the coefficient values c(n)
are given by:
c(n) =S(n) ·w(n) (3.27) In this case S(n) corresponds to the sinc function impulse response of the ideal filter, and w(n) corresponds to the applied Kaiser-Bessel window. For the two band-edge frequencies Fa and Fb and a sample frequency Fs,
3.3 Modulation of the interferometer 29
Figure 3.9: A typical transfer function of a digital low-pass (FIR) filter. Note the side lobes, which arise due to the fact that if one averages at certain frequencies, the result will automatically be zero. The FIR filter is used to demodulate the signal obtained in figure 3.7.
S(n) is given by: S(n) = sin(2πn Fb Fs) −sin(2πn Fa Fs) πn (3.28) for 0≤n<N. w(n) is given by: w(n) = I0(α q 1− (n−MM)2) I0(α) (3.29)
for 0 ≤ n < N. Here N is the order of the filter. To do his job properly, the order of the filter should be at least equal to D−8
14.36d fFs. M is just N−1
2 . Here D is the required attenuation after the transition band (given in dB) df is the with of the transition band and Fa and Fb are the two band edge frequencies. Fsis the sample frequency. For a low-pass or a high-pass filter,
3.4 Differentiate and cross multiply method 30
Faand Fb may be considered to be zero respectively. I0 is the zeroth order Bessel function of the first kind. α is the so-called Kaiser-Bessel window shape factor. The required window shape factor is given by 0.5842(D−
210.4) +0.07886(D−21)if the required attenuation is smaller than 50db. If the required attenuation is larger, this formula changes to 0.1102(D−8.7). [9]
Complex phase plotting
With the obtained quadrature signals, a phasor diagram can be created. The resulting diagram plots sω against s2ω, which should form an ellipse, with an offset formed by sDC. This offset value can, if one prefers, be filtered out.
3.4
Differentiate and cross multiply method
The arctangent method described above is not the commonly used ap-proach because it requires fast processing. The commonly used method is called DCM (differentiate-and-cross-multiply). As in the method de-scribed above, in DCM at first the signal is multiplied with a sin ωt and a sin 2ωt function. Hereafter, both signals are lowpass-filtered. The fil-tered signals are then multiplied by the time derivative of the other sig-nals, which is typically done in an analog manner. This should lead to a similar phasor diagram. However, if the motor is not moving for a while, the derivative will go to zero and the output will start to drift. Therefore, it is preferred to use the arctangent method.Chapter
4
Experiments and Results
4.1
Introduction
As mentioned earlier, the goal of this project is to further develop the cryo-LEEM setup, or more specifically, to determine the feasibility and the effi-ciency of using an interferometer to measure the position of the piezomo-tors. Something that affects the feasibility is the possibility to align the fiber and the mirror. Therefore it is necessary to determine the misalign-ment angle where the setup stops working. It also makes sense to investi-gate whether the modulation method works as expected. Thereafter, one could try to test the setup and investigate whether the obtained results are as expected, to eventually conclude if the method of using a fiber interfer-ometer is a feasible and efficient thing to do.
4.2
Setup
4.2.1
Optics
The setup is constructed on an optical table. As said, fibers are used to guide the light. The light source which is used is a laser diode, with a wavelength of 1534 nm. The laser beam is reflected by a mirror. The mir-ror which is used has a reflectivity of about 95%. It is possible to tilt the mirror in two directions using a mirror mount. This mirror mount has a resolution of 8 mrad per knob cycle. The used mirror stage enables the mirror to move forward and backwards, in a range of about one cm. The laser diode has a current limit of 50 mA and at the maximal power it re-mains a class 1 laser. The laser emits light directly into a fiber, forming an
4.2 Setup 32
arm of the fiber splitter. The fiber splitter divides the light into two fibers. The first fiber is the reference arm. Using this arm, one could choose the type of interferometer one would like to use. In the Michelson type, this fiber leads to a retroreflector, which reflects the light back. If one would like to use the Fabry-Perot type interferometer, this arm is not used. In that case, one replaces the retroreflector by an beam dump. The dump absorbs the entering light. The other fiber leads to the collimating lens. This could be either a GRIN-lens or a normal collimator. A GRIN-lens (Gradient In-dex Lens) is basically a tube made of glass which works as a converging lens due to the increasing refractive index at a larger radius. The collima-tor is fixed in a lens holder. To make it easier to switch between the lenses, a lens stage has been designed and built. which can be screwed onto the optical table. The lens focusses light on the mirror. The distance between lens and mirror can be modified using the stage to which the mirror is at-tached This stage uses a piezomotor to move the mirror and measure the displacement. The light reflected from the mirror is coupled back into the fiber, and goes back to the fiber splitter into the fourth arm, which leads to a photodiode. To connect the laser, the fiber splitter and the photodiode, fiber couplers are used. The mirror stage, the collimating lens standard and the fiber couplers are fixed on the optical table using screws. The entire setup is shown in figure 4.1. The devices controlling the setup are shown in figure 4.2.
4.2.2
Electronics
To construct the electronical setup of the laser and the photodiode, a bread-board is used, Although the breadbread-board has its own power supply, an-other power supply is used, because the supply of the breadboard is found to be too noisy. The laser source is driven by the function generator. The voltage inputs are connected to a summing amplifier. The laser source is a VI-converter. A simplified schematic picture is shown in figure 4.3.
The photodiode is connected to the power supply in the reverse-bias mode. The power supply is set at 8V. The strength of the current the pho-todiode generates depends on the irradiance of the incoming light. It is necessary to convert the current into a voltage to be able to use the DAQ-device, The computer needs the DAQ device to obtain the data. Therefore the current goes through an IV converter to generate a voltage. This IV converter uses a resistance of 2 kΩ (see figure 4.4). This resistance can be used to calculate the power, (byVR2) which corresponds to the irradiance, since the irradiance is the power per unit area.
4.2 Setup 33
Figure 4.1: The interferometer setup used during the experiment. In the rear, en electronic breadboard can be seen. The little blue box right next to the bread-board is the DAQ-device. The fiber splitter is located in the center. The mirror is attached to a piezomotor, which is controlled using the green control unit at the front. The lens standard is placed behind the mirror.
4.2.3
Software
The used computer controlled the DAQ-device using the python program-ming language, extended with two additional programs, DAQmx and Py-DAQmx. DAQmx is the usual program to control this kind of devices. However, the device is designed to be controlled using another program-ming language, named Labview, which is developed by the company that also developed the DAQ device itself. But since in this experiment it is pre-ferred to use python, it was also needed to install the program PyDAQmx, which acts as a kind of a translator. The DAQ-device is connected to a computer with an USB-cable.
4.2 Setup 34
Figure 4.2: The devices controlling the interferometer setup. At the far left, one finds the backside of the optical table. Next to the optical table, there are three power supplies. The devices next to the power supplies are, from bottom to top, the two lock-in amplifiers, the function generator and an oscilloscope. The data processing computer can be found on the right.
4.2 Setup 35
Figure 4.3: A simplified circuit diagram of the laser control circuit. There are two 0-10V inputs, one for the DC setpoint and one for the modulation signal. These are combined by a summing amplifier and converted to a current by a V-I converter.
Figure 4.4:The simplified circuit diagram of the photodiode amplifier. The diode is connected to a power supply of 10V and, since the photodiode generates a current instead of a voltage, to an IV-converter to enable the DAQ device, which is connected after the IV-converter, to process the data.
4.3 Measurements 36
4.3
Measurements
4.3.1
Determining setup characterestics
In order to determine the characteristics of the setup,first, the power of the laser has been measured as a function of the current. The result of that measurement is given in figure 4.5. The intensity at different parts of the fibersplitter is also measured, this was done by replacing the photodiode instead of the lens or the attenuator. The result is shown in figure 4.6. It is observed that attaching the reflector leads to a reflectivity of about 90%, whereas attaching the attenuator almost removes the reflection.(to 0.1%.)
Figure 4.5: The power diagram as a function of the current. The current is
mea-sured by measuring the voltage over a 200 Ω resistor. Note that above the
so-called threshold current of 20mA, the diagram approximately corresponds to a straight line. This is useful because the intensity is modulated as well, because of the linearity of this curve, this intensity modulation can be compensated in software.
4.3 Measurements 37
Figure 4.6:The result of the relative power measurements in the entire fibersplit-ter. The values show the fraction of light compared to the light coming directly from the laser. The figures correspond to the case of a bare fiber end. The direc-tion of the lightpaths is the direcdirec-tion at which the number is at the right side of the fiber. One could notice a weak reflection at the fiber end.
4.3.2
Angle Dependence measurement
This measurement is performed by measuring the voltage as a function of the misalignment of the mirror. The interferometer was in the Fabry Perot mode, which means that the GRIN-lens is used in combination with the beam dump. It is also attempted to form an improved lens. The new lens consisted of a commercially available GRIN-lens, on which a layer, with a thickness of 1 nm, of chromium is sputtered. The objective of doing this is to increase the reflectivity of the lens, to investigate whether this would result in a lower angle sensitivity. The coated lens is put in a tube made of glass, together with a ’pigtailed ferrule’. A ’pigtailed ferrule’ is a piece of glass which is attached to a fiber wire. The tube is put in a hole in a new lens holder, which could be attached to the standard. Then the irradiance is measured as a function of the misalignment angle again. The results of this test are shown in figure 4.7:
4.3 Measurements 38
Figure 4.7: The result of the angle dependence measurements for the coated GRIN-lens. The figure compares the coated lens with a normal collimator (shown in green). A gaussian fit is drawn to fit the points. Note that the GRIN-lens is eas-ier to align, but the maximal intensity is lower.
4.3.3
The modulated signal
The modulation was done by varying the current of the laser setup using a function generator. The function generator used an offset voltage of 6 V. The amplitude is varied from 0.6 to 2 volts, which, using a wavelength change of 4 pm/mA, should correspond to wavelength changes from 12 -40 pm. The modulated signal is shown in figure 4.8.
4.3.4
Phasor diagram
In order to demodulate the modulated signal, two lock-in amplifiers were used. The Lock-in amplifiers were connected to the oscilloscope to visual-ize the measurement. The oscilloscope was set in the xy-mode to be able to show the value point of the phasor diagram. If this principle works well, the point should move along an ellipse, and after it has completed one
ro-4.3 Measurements 39
Figure 4.8:The measured modulated signal. The signal looks different compared to the theoretical modulated function in figure 3.7, because the correction for the intensity modulation has not been applied yet.
tation, the change in path length will be equal to half the wavelength of the light. The center of the ellipse is probably not at the origin. This is due to the effect that modulation of the laser also results in intensity fluc-tuations. One could make use of a program that modifies the center, the inclination angle and the axes ratio so that the ellipse is projected on a unit circle, which will make it easier to observe the phase.
4.3.5
Interferometry without a lens.
One could already do a measurement on the fiber coupler if both the lens and the reflector (or the attenuator) are even disconnected. In that case, one has a fiber coupler with two bare fiber ends at one side. If one changes the temperature of a fiber end (This could be done by, for example, using a hand to hold the fiber.), the length of that fiber end will change due to thermal expansion and therefore one still may observe an interference pattern. The signal coming from this setup is weaker, because only 5% of the power is reflected, therefore the signal is amplified by changing the gain of the IV converter. Then one could warm up the fiber, and if the fiber end is released, measure the interference pattern until the fiber end has cooled down again. The measurement of the intensity (using an oscilloscope) is given in figure 4.10. Hereafter one could use the entire setup and plot a phasor diagram. This was done with various modulation depths. The result is given in figure 4.11. The ratio of the long and the short axis of the plotted ellipse of the phasor diagram is also measured. The result is given in figure 4.9. The final measurements of the length differences is given in figure 4.12.
4.3 Measurements 40
Figure 4.9: The axes lengths of the measured ellipses for various modulation depths. One can see that for higher modulating voltages, the axes lengths are quite equal. Therefore the J1/J2crossover at β=2.6 is expected to be there.
Figure 4.10: The measured signal after releasing one fiber end after it was held by hand for a few seconds. One can observe the interference pattern due to the decreasing length of the fiber end. One can also observe the increase in period of the signal, due to the decrease in the cooling rate.
4.3 Measurements 41
Figure 4.11:A quadrature plot, measured with the setup with two bare fiber ends. The ellipse has a small inclination, which arises from the fluctuating intensity. (The intensity change was not corrected)
4.3.6
GRIN-lens measurements
The GRIN-lens is connected again to perform the real position measure-ments. The measurements were done at various distances between the stages of the mirror and the lens, to observe whether this would affect the J2−J1ratio. A result is given in figure 4.13. It turned out that quite a sen-sitive setup was built. A position value change of 10λ is once achieved by the heat of an approaching hand at several centimeters. In other words, the Michelson interferometer is not very stable and it is sensitive to things like temperature fluctuations, external noise influences and basically ev-erything else. Nevertheless, that is actually very useful for testing the setup. The results of the position measurement plot using a written pro-gram. One could observe the measured ellipse, as well as the ellipse trans-formed into a unit circle by correcting the center of the origin, the ratio of the axes, and the angle of inclination. The unit circle is divided into eight octants, in order to monitor the number of complete cycles to be able to measure the position outside the range of one cycle. In an additional plot the displacement from the start of the measurement is displayed. These plots are shown in figure 4.14.
4.3 Measurements 42
Figure 4.12: The measured result of changing the arm length. In the top left plot, one can observe the measured quadrature signal. The quadrature signal is automatically corrected with an online algorithm, to a unit circle at the top right plot. The phasor diagram plots are reset if a full cycle is completed. At the bottom plot, the displacement is plotted against the time in seconds. The length difference is given in micrometers.
4.3 Measurements 43
Figure 4.12:In this plot one could observe the heating of one of the fiber ends, af-ter heating this fiber end that end is cooled again in combination with heating the other fiber. Note that increasing the length of one and the other fiber corresponds to positive and negative length differences respectively.
4.3 Measurements 44
Figure 4.13: A quadrature plot, measured with the Michelson setup (using a re-flector and the GRIN-lens). One could again observe the inclination angle of the ellipse. This result seems to have more distortions compared to the setup without a lens. These are caused by imperfections of the reflector, and spurious reflections in the fiber couplers.
Figure 4.14: The result of the position measurement of the mirror. The mirror is attached to the piezomotor. The position of the motor is varied using the control device shown in figure 4.1. The position is given from the start of the measure-ment, up to 200 seconds. The position is given in micrometers. The interferometer position corresponded to the position indicated by the sensor in the piezomotor.
4.4 Discussion 45
4.3.7
Digital lock-in
The device that is built worked according to the expectations, but the device consisted of two large, heavy lock-in amplifiers, which made the device not very useful for position measurement in the LEEM. Since the DAQ-device is also able to generate a voltage, it is attempted to create a digital lock-in machine. using the DAQ-device. The digital lock-in worked according to the same principle as the original lock-in. The DAQ device sent out the constant offset voltage, as well as the modulating voltage. The incoming values were stored in two lists, where they are multiplied by the sine wave and low-pass filtered using a FIR-filter to substitute the lock-in amplifiers. Then the values in the two lists can be plotted to form a phasor diagram.
4.4
Discussion
At small modulation depths (small distances) the quadrature plot is an ellipse. The intensity modulation that is associated with the frequency modulation of a low cost DFB laser skews this ellipse. An algorithm has been developed that performs the correction of this signal to a unit cir-cle on line. In the angle dependence measurements, the improved GRIN-lens seemed to be really an improvement regarding the fringe contrast, but then one should consider that the signal is much weaker. This can par-tially be solved by further amplifying the signal with a low noise amplifier. However, this setup is still sensitive to spurious reflections in the fiber: in figure 4.8, the picture of the modulated signal, there are some small fluc-tuations visible at the top of the peaks, which are a distortion of the signal. It also turned out that the fiber surfaces should be as clean as possible, since already tiny impurities are observed to bring distortions. Further-more, the FC-PC fiber connectors present in the setup should be replaced by APC connectors. In this kind of connectors, the fiber is cleaved under an angle of 8 degrees. This means that the effect of internal reflections is reduced. It might also be good to know that the laser has its own built-in photodiode. One could use this photodiode to correct for the built-intensity fluctuations which arise if one applies frequency modulation, by dividing the signal with the measured intensity.
Chapter
5
Conclusion
Based on the obtained results, it is concluded that using an interferometer is a feasible and efficient way to measure a position, also in ultra high vac-uum at low temperatures. Therefore next steps in this project are about to be taken. The most important step is to use a so-called cube beamsplitter, cemented onto the collimator. A cube beamsplitter is made of 2 prisms that are cemented together with a 50% reflecting mirror on the diagonal. The reflected part of the light is projected on another mirror, coated on the side of the cube. The transmitted part goes to the original mirror. In fact, the cube beamsplitter is very similar compared to a Michelson interferom-eter. The main advantage is that 50% of the light is used for measurements, instead of 5% in the Fabry-Perot-setup. At the same time, one uses a very short reference arm, which is insensitive to external influences, because to first order, thermal drift is eliminated. In fact the cube beamsplitter com-bines the advantages of the Michelson and the Fabry-Perot setup. The setup including the cube beamsplitter is given in figure 5.1. Finally, efforts are on the way to replace the lock-in amplifiers by an FPGA-driven control system. With these modifications, a relatively simple, low-cost and accu-rate position sensor is obtained, which can be used to measure the position of the piezomotors at the cryogenic side of the LEEM.
47
Figure 5.1: The setup of the interferometer in combination with the cube beam-splitter. The cubesplitter is used in combination with the attenuator. The red line is the light coming out of the Grinlens. after transmitting the beamsplitter the intensity is half as large, expressed by the pink line.
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